Unimodular Hyperbolic Triangulations: Circle Packing and Random Walk

Abstract

We show that the circle packing type of a unimodular random plane triangulation is parabolic if and only if the expected degree of the root is six, if and only if the triangulation is amenable in the sense of Aldous and Lyons. As a part of this, we obtain an alternative proof of the Benjamini-Schramm Recurrence Theorem. Secondly, in the hyperbolic case, we prove that the random walk almost surely converges to a point in the unit circle, that the law of this limiting point has full support and no atoms, and that the unit circle is a realisation of the Poisson boundary. Finally, we show that the simple random walk has positive speed in the hyperbolic metric.

40 pages, 5 figures; minor revision, new and improved proof for Proposition 1.2

References in corpus (3)

• Boundaries of planar graphs, via circle packings
• Planar stochastic hyperbolic infinite triangulations
• Anchored expansion, speed, and the hyperbolic Poisson Voronoi tessellation

Cited by in corpus (6)

• Mating of trees for random planar maps and Liouville quantum gravity: a survey
• Boundaries of Planar Graphs: A Unified Approach
• Invariant embeddings of unimodular random planar graphs
• Some criteria for circle packing types and combinatorial Gauss-Bonnet Theorem
• On the site percolation threshold of circle packings and planar graphs
• A full characterization of invariant embeddability of unimodular planar graphs